Suppose that $G$ is a simple, undirected graph with $n$ vertices and $m$ edges. Conjecture: The total number of vertex paths of length $l$ is at most $$ n (2 m/n)^{l-1} $$

The heuristic basis for this is that the average degree is $2 m/n$, and this is the total number of ways of choosing paths $l$ vertices in such a graph.

Is this conjecture true? Is there another bound known on the number of such paths?